Pluripotential theory on Teichm\"uller space I -- Pluricomplex Green function

TL;DR

This paper explores pluripotential theory on Teichmüller space, providing an alternative proof of the pluricomplex Green function formula and revealing its stratified structure and Levi form description.

Contribution

It introduces a new approach to the Krushkal formula and describes the stratified real-analytic structure and Levi form of the pluricomplex Green function on Teichmüller space.

Findings

Alternative approach to Krushkal formula

Stratified structure of Teichmüller space

Levi form description via Thurston symplectic form

Abstract

This is the first paper in a series of investigation of the pluripotential theory on Teichm\"uller space. The main purpose of this paper is to give an alternative approach to the Krushkal formula of the pluricomplex Green function on Teichm\"uller space. We also show that Teichm\"uller space carries a natural stratified structure of real-analytic submanifolds defined from the structure of singularities of the initial differentials of the Teichm\"uller mappings from a given point. We will also give a description of the Levi form of the pluricomplex Green function using the Thurston symplectic form via the Dumas symplectic structure on the space of holomorphic quadratic differentials.